Download Z-scores and the Standard Normal Table Z-scores

Z-scores and the Standard Normal Table
Z-scores are a way to make difficult comparisons of values from different locations,
times, or situations. The z-score counts up a number of standard deviations from the
mean.
Here is an example –
Bob gets a 88% in Mr. Klump’s Chemistry class. The average in the class is 80% with a
standard deviation of 6%.
Sue gets a 92% in Mrs. Quark’s Chemistry. The average in this class is 82% with a
standard deviation of 10%.
Who did better? Calculate a z-score z =
x−x
. This formula says “Take the score,
σ
minus the mean. Then divide by the standard deviation. Think of a z-score as how many
standard deviations from the mean a given score is.
In this example Bob’s z-score is z =
88 − 80 8 4
= = ≈ 1.33 . Sue’s z score is
6
6 3
92 − 82 10
=
= 1.00 . Even though Sue had the higher average, Bob’s z-score is
10
10
higher. We can say that Bob actually did better as long as the courses had students of the
same ability and in most other respects were the same.
z=
We can also talk about the probabilities of getting these scores randomly using the table
of standard normal probabilities distributed in class. This distribution is a cumulative
probability distribution based on z-scores. To use this table, calculate the z-score and
look up the probability of randomly getting less than that amount in the body of the table.
In this example for Bob P(x≤88) = P(x≤ 1.33) = .9082. For Sue P(x≤ 92)=P(z≤ 1.00) =
.8413
One way to interpret these numbers is that Bob scored higher than about 90% of his class
and Sue scored higher than 84% of hers. This is more evidence that Bob did better.
For each of the following problems use these values for a 150 point final test: mean = 83
and standard deviation of 16. Calculate z-score(s), draw a normal curve with an
appropriate shaded region, and find the probability. Show all work on graph paper.
1. What is the probability of getting less than 83?
2. What is the probability of getting less than 67?
3. What is the probability of getting more than 67?
4. What is the probability of getting less than 99?
5. What is the probability of getting more than 99?
6. What is the probability of getting between 67 and 99?
7
What is the probability of getting less than 80?
8 What is the probability of getting more than 110?